Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Armenian Journal of Physics

Numerical Solutions of Certain Time-Independent Quantum Mechanics Problems by Using the Python Environment

H. A. Shahinyan, L. Ghazaryan, Arsen Babajanyan

Published Dec 20, 2024• 0 citations
Publisher

Abstract

This paper discussed the digitization of some analytical solutions of the fundamental equation of non-relativistic quantum mechanics, the Schrödinger equation, in the Python programming environment. The main objective of the work is to automate the solution of several known analytically solvable problems of quantum mechanics and ensure their use in more complex problems. For this purpose, the Python programming environment was chosen, which, due to the large number of libraries and flexibility, is currently widely used in solving physical and mathematical problems. In the program we propose, which is available on the GitHub platform, it is solved easily. The problem of determining the probability of the spatial distribution of an electron in a hydrogen-like atom is discussed. The Schrödinger equation is presented in the spherical coordinate system. The wave function, describing the electron's state, quantized energy values, and probability density, is derived analytically. A recursive function was written in Python to handle the series coefficients defining the wave function. Utilizing relevant libraries, the resulting program constructs electron spatial distribution functions for different excitation levels. The paper then examines two one-dimensional tunneling scenarios and provides the tunneling coefficients in analytical form. A Python program was developed to plot the dependence of these coefficients on the particle's coordinate and energy. All the code is available on GitHub.

DOI
10.54503/18291171-2024.17.3-43

Related papers

Visualisasi Komputasi Fungsi Gelombang dan Probabilitas Partikel Kuantum dalam Sumur Potensial Tak Hingga Satu Dimensi

Faizah Istisania, Vina Yanti, Hamdi Akhsan +1 more

This research aims to develop computational visualization of the wave function and probability distribution of particles in a one dimensional infinite potential well. The wave function is obtained analytically from the time-independent Schrödinger equation and then normalized to give it probabilistic meaning. Numerical simulations are carried out using Python with the NumPy library for computation and Matplotlib for visualization, allowing manipulation of parameters such as box length and quantum number. The visualization results confirm the basic concepts of quantum mechanics, namely quantized energy levels, sinusoidal wave function forms with a specific number of nodes, and probability distributions that have several peaks and are always positive. The simulation also shows the correspondence between wave-function nodes and zero-probability points, as well as the increasing complexity of oscillations and variations in probability distributions as the quantum number increases. These findings are consistent with theoretical predictions and show that the computational approach is effective in visualizing abstract concepts in quantum mechanics, thereby clarifying the relationship between mathematical formulation and its physical interpretation.

Jurnal Teori dan Aplikasi Fisika•Jul 31, 2026•0 citations

Solution of Time-Independent Schrodinger Equation (TISE) by Using Finite Difference Approach

I Gusti Agung Widagda, Nengah Artawan, I Gde Antha Kasmawan +2 more

This research investigates the numeric solution of the time-independent Schrödinger equation for the quantum harmonic oscillator by finite difference approach. The harmonic oscillator, described by a quadratic function potential, is a fundamental model in quantum mechanics due to its broad applications, ranging from molecular vibrations to quantum field theory. The time-independent Schrödinger equation is a second-order differential equation that typically poses challenges when solved analytically for complex potentials. The finite difference method become an attractive choice as it transforms the continuous differential equation into a system of linear equations that can be computationally solved through computer programming code. In this study, the spatial domain is discretized, and the second derivative is calculated by using central differences, transforming the TISE into a tridiagonal matrix representing Hamiltonian of system. By finding solutions to this matrix eigenvalue problem, wavefunctions and eigenvalues are obtained. The study results demonstrate that the finite difference approach effectively solves the TISE for the harmonic oscillator. The results obtained by using the finite difference method closely approximate the analytical results. The linear regression results show respectively that the gradient (β1), regression coefficient (β0) and coefficient of determination (R²) approach ideal values of 1, 0, and 1. The z-test results also show that the value of calculated z < critical z, indicating that the wavefunction and probability density, whether estimated by using finite difference approach or analytical methods, are equivalent with confidence level of 95 percent.

BULETIN FISIKA•Feb 28, 2025•0 citations

Visualization of Probability Distribution for a Particle in a One-Dimensional Box using Computational Simulation

Zahra Ayanna Firmansyah, Nur Kayla Lekati, Aisyah Veronika +2 more

This study presents a computational visualization of the stationary states of a particle confined in a one-dimensional infinite potential well, formulated entirely from analytical solutions and implemented in a cloud-based Python environment. Using Google Colab, wave functions, probability densities, and nodal structures were generated for several quantum numbers to illustrate fundamental quantum-mechanical characteristics, including energy quantization, spatial oscillatory behavior, and the emergence of classical correspondence at higher energy levels. The resulting visualizations offer a clear representation of the spatial distribution of quantum states, thereby supporting a conceptual understanding of the underlying analytical model. As a novelty, this study introduces a Colab-based analytical–numerical combination as a visualization medium, a format rarely explored in depth in quantum-well learning contexts. Beyond instructional applications, the modeled system accurately reflects the essential physical behavior of electrons in semiconductor quantum wells, where quantized subband energies and wave-function profiles significantly influence optical and electronic transitions. These findings demonstrate that analytically derived quantum models, when integrated with computational visualization, can effectively enhance conceptual comprehension while offering insight into the fundamental principles governing modern optoelectronic structures.

Kappa Journal•Dec 24, 2025•0 citations

The Laplace Method for Energy Eigenvalue Problems in Quantum Mechanics

Jeremy Canfield, Anna Galler, James K. Freericks

Quantum mechanics has about a dozen exactly solvable potentials. Normally, the time-independent Schrödinger equation for them is solved by using a generalized series solution for the bound states (using the Fröbenius method) and then an analytic continuation for the continuum states (if present). In this work, we present an alternative way to solve these problems, based on the Laplace method. This technique uses a similar procedure for the bound states and for the continuum states. It was originally used by Schrödinger when he solved the wave functions of hydrogen. Dirac advocated using this method too. We discuss why it is a powerful approach to solve all problems whose wave functions are represented in terms of confluent hypergeometric functions, especially for the continuum solutions, which can be determined by an easy-to-program contour integral.

arXiv:2208.07433•Quantum Reports•Apr 20, 2023•3 citations

The Analytical Solutions of the Schrodinger Equation for a Single Electron in the Nikiforov-Uvarov Framework

Yacobus Yulianto, Zaki Su’ud

In this study, it is intended to show an alternative method to derive the wave function of a single electron as solutions of the Schrodinger equation. The Nikiforov-Uvarov method was chosen to be utilized since this method can solve the Schrodinger equation with several well-known potentials in the non-relativistic mechanics of quantum. The obtained results of this study have succeed to explain the wave function and the energy eigenvalue for a single electron as lectured in quantum physics textbooks. These results prove that the Nikiforov-Uvarov method provides an alternative procedure to solve the Schrodinger equation.

Jurnal Fisika•Nov 23, 2020•1 citation

Modeling Time-Independent Schrödinger Equation for an Infinite Potential Well Using Numerov and Matrix Methods

Dalal Y. Saad, Fawzi A. Ikraiam

The objective of this study is to numerically solve and apply two approximate methods to investigate the Time Independent Schrödinger Equation (TISE) in one dimension for an infinite potential square well. These two numerical methods are the Numerov Method (NM) and Matrix Method (MM). As a simulation tool, MATLAB, a high-level programming language and an efficient simulation tool, is used for modeling and solving TISE in one dimension. Exact analytical solutions for these potential functions are obtained and compared with numerical solutions and computational techniques. The energy eigenvalues and Eigen functions of a particle (such as an electron) restricted to move inside this potential are discussed as an illustration. The numerically calculated energies of several states with increasing numbers of points were obtained from both methods and compared with the simulation results of the exact solution. As an exemplary case, the first five wave functions are accurately determined numerically where the discreteness is found since the wave function vanishes at the boundary. The obtained results show very good agreement and the similarity is clearly confirmed between the three cases. This agreement confirms that this approach was highly accurate and efficient. The accuracy and the convergence of the numerical obtained results were easily checked. The stability of these methods is due to the fact that there are no restrictions on the time steps to be taken. The merits of these numerical methods are to avoid a huge expense in time when solving Schrödinger equation.

Libyan Journal of Science &Technology•Mar 1, 2025•0 citations
This paper discussed the digitization of some analytical solutions of the fundamental equation of non-relativistic quantum mechanics, the Schrödinger equation, in the Python programming environment. The main objective of the work is to automate the solution of several known analytically solvable problems of quantum mechanics and ensure their use in more complex problems. For this purpose, the Python programming environment was chosen, which, due to the large number of libraries and flexibility, is currently widely used in solving physical and mathematical problems. In the program we propose, which is available on the GitHub platform, it is solved easily. The problem of determining the probability of the spatial distribution of an electron in a hydrogen-like atom is discussed. The Schrödinger equation is presented in the spherical coordinate system. The wave function, describing the electron's state, quantized energy values, and probability density, is derived analytically. A recursive function was written in Python to handle the series coefficients defining the wave function. Utilizing relevant libraries, the resulting program constructs electron spatial distribution functions for different excitation levels. The paper then examines two one-dimensional tunneling scenarios and provides the tunneling coefficients in analytical form. A Python program was developed to plot the dependence of these coefficients on the particle's coordinate and energy. All the code is available on GitHub.
This research aims to develop computational visualization of the wave function and probability distribution of particles in a one dimensional infinite potential well. The wave function is obtained analytically from the time-independent Schrödinger equation and then normalized to give it probabilistic meaning. Numerical simulations are carried out using Python with the NumPy library for computation and Matplotlib for visualization, allowing manipulation of parameters such as box length and quantum number. The visualization results confirm the basic concepts of quantum mechanics, namely quantized energy levels, sinusoidal wave function forms with a specific number of nodes, and probability distributions that have several peaks and are always positive. The simulation also shows the correspondence between wave-function nodes and zero-probability points, as well as the increasing complexity of oscillations and variations in probability distributions as the quantum number increases. These findings are consistent with theoretical predictions and show that the computational approach is effective in visualizing abstract concepts in quantum mechanics, thereby clarifying the relationship between mathematical formulation and its physical interpretation.
This research investigates the numeric solution of the time-independent Schrödinger equation for the quantum harmonic oscillator by finite difference approach. The harmonic oscillator, described by a quadratic function potential, is a fundamental model in quantum mechanics due to its broad applications, ranging from molecular vibrations to quantum field theory. The time-independent Schrödinger equation is a second-order differential equation that typically poses challenges when solved analytically for complex potentials. The finite difference method become an attractive choice as it transforms the continuous differential equation into a system of linear equations that can be computationally solved through computer programming code. In this study, the spatial domain is discretized, and the second derivative is calculated by using central differences, transforming the TISE into a tridiagonal matrix representing Hamiltonian of system. By finding solutions to this matrix eigenvalue problem, wavefunctions and eigenvalues are obtained. The study results demonstrate that the finite difference approach effectively solves the TISE for the harmonic oscillator. The results obtained by using the finite difference method closely approximate the analytical results. The linear regression results show respectively that the gradient (β1), regression coefficient (β0) and coefficient of determination (R²) approach ideal values of 1, 0, and 1. The z-test results also show that the value of calculated z < critical z, indicating that the wavefunction and probability density, whether estimated by using finite difference approach or analytical methods, are equivalent with confidence level of 95 percent.
This study presents a computational visualization of the stationary states of a particle confined in a one-dimensional infinite potential well, formulated entirely from analytical solutions and implemented in a cloud-based Python environment. Using Google Colab, wave functions, probability densities, and nodal structures were generated for several quantum numbers to illustrate fundamental quantum-mechanical characteristics, including energy quantization, spatial oscillatory behavior, and the emergence of classical correspondence at higher energy levels. The resulting visualizations offer a clear representation of the spatial distribution of quantum states, thereby supporting a conceptual understanding of the underlying analytical model. As a novelty, this study introduces a Colab-based analytical–numerical combination as a visualization medium, a format rarely explored in depth in quantum-well learning contexts. Beyond instructional applications, the modeled system accurately reflects the essential physical behavior of electrons in semiconductor quantum wells, where quantized subband energies and wave-function profiles significantly influence optical and electronic transitions. These findings demonstrate that analytically derived quantum models, when integrated with computational visualization, can effectively enhance conceptual comprehension while offering insight into the fundamental principles governing modern optoelectronic structures.
Quantum mechanics has about a dozen exactly solvable potentials. Normally, the time-independent Schrödinger equation for them is solved by using a generalized series solution for the bound states (using the Fröbenius method) and then an analytic continuation for the continuum states (if present). In this work, we present an alternative way to solve these problems, based on the Laplace method. This technique uses a similar procedure for the bound states and for the continuum states. It was originally used by Schrödinger when he solved the wave functions of hydrogen. Dirac advocated using this method too. We discuss why it is a powerful approach to solve all problems whose wave functions are represented in terms of confluent hypergeometric functions, especially for the continuum solutions, which can be determined by an easy-to-program contour integral.
In this study, it is intended to show an alternative method to derive the wave function of a single electron as solutions of the Schrodinger equation. The Nikiforov-Uvarov method was chosen to be utilized since this method can solve the Schrodinger equation with several well-known potentials in the non-relativistic mechanics of quantum. The obtained results of this study have succeed to explain the wave function and the energy eigenvalue for a single electron as lectured in quantum physics textbooks. These results prove that the Nikiforov-Uvarov method provides an alternative procedure to solve the Schrodinger equation.
The objective of this study is to numerically solve and apply two approximate methods to investigate the Time Independent Schrödinger Equation (TISE) in one dimension for an infinite potential square well. These two numerical methods are the Numerov Method (NM) and Matrix Method (MM). As a simulation tool, MATLAB, a high-level programming language and an efficient simulation tool, is used for modeling and solving TISE in one dimension. Exact analytical solutions for these potential functions are obtained and compared with numerical solutions and computational techniques. The energy eigenvalues and Eigen functions of a particle (such as an electron) restricted to move inside this potential are discussed as an illustration. The numerically calculated energies of several states with increasing numbers of points were obtained from both methods and compared with the simulation results of the exact solution. As an exemplary case, the first five wave functions are accurately determined numerically where the discreteness is found since the wave function vanishes at the boundary. The obtained results show very good agreement and the similarity is clearly confirmed between the three cases. This agreement confirms that this approach was highly accurate and efficient. The accuracy and the convergence of the numerical obtained results were easily checked. The stability of these methods is due to the fact that there are no restrictions on the time steps to be taken. The merits of these numerical methods are to avoid a huge expense in time when solving Schrödinger equation.